Dog story being read out (4)
I believe the answer is:
tail
'dog' is the definition.
('tail' can be a synonym of 'dog')
'story being read out' is the wordplay.
'story' becomes 'tale' ('tale' can be a synonym of 'story').
'being read out' indicates a 'sounds like' (homophone) clue.
'tale' is a homophone of 'TAIL'.
(Other definitions for tail that I've seen before include "Follow furtively like private detective" , "Follow - train" , "Follow closely - taper (off)" , "Private investigator" , "Follow the caudal appendage" .)